54edo: Difference between revisions
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== Theory == | == Theory == | ||
54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This scale is so [[soft]], with {{nowrap|L/s {{=}} 8/7}}, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]]. | 54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This scale is so [[soft]], with {{nowrap|L/s {{=}} 8/7}}, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]]. | ||
The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest edo in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600{{c}} apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. | The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest edo in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600{{c}} apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. | ||
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=== Octave stretch === | === Octave stretch === | ||
54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[ | 54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[38ed5/3]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1. | ||
If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1. | If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1. | ||
[[ | [[40ed5/3]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]]. | ||
There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings. | There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings. | ||
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[[Category:Todo:add rank 2 temperaments table]] | [[Category:Todo:add rank 2 temperaments table]] | ||
== Music == | |||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/Bi5-YQUQHek ''microtonal improvisation in 54edo''] (2025) | |||